Nonlinear Differential Equations and Applications (NoDEA)最新文献

We investigate the Pompeiu property for subsets of the real line, under no assumption of connectedness. In particular we focus our study on finite unions of bounded (disjoint) intervals, and we emphasize the different results corresponding to the cases where the function in question is supposed to have constant integral on all isometric images, or just on all the translation-images of the domain. While no set of the previous kind enjoys the Pompeiu property in the latter sense, we provide a necessary and sufficient condition in order a union of two intervals to have the Pompeiu property in the former sense, and we produce some examples to give an insight of the complexity of the problem for three-interval sets.

We prove an integral representation result for a class of variational functionals appearing in the framework of hierarchical systems of structured deformations via a global method for relaxation. Some applications to specific relaxation problems are also provided.

In the present paper, we study the fast rotation and inviscid limits for the 2-D dissipative surface quasi-geostrophic equation with a dispersive forcing term, in the domain (Omega =mathbb {T}^1times mathbb {R}). In the case when we perform the fast rotation limit (keeping the viscosity fixed), in the context of general ill-prepared initial data, we prove that the limit dynamics is described by a linear equation with parabolic structure. Conversely, performing the combined fast rotation and inviscid limits, we show that the means of the target initial datum (overline{vartheta }_0) are conserved along the motion. The proof of the convergence is based on a compensated compactness argument which allows, on the one hand, to get compactness properties for suitable quantities hidden in the wave system and, on the other hand, to exclude the oscillatory part of waves at the limit.

We prove that the classical hyperbolic Kirchhoff equation admits global-in-time solutions for some classes of initial data in the energy space. We also show that there are enough such solutions so that every initial datum in the energy space is the sum of two initial data for which a global-in-time solution exists. The proof relies on the notion of spectral gap data, namely initial data whose components vanish for large intervals of frequencies. We do not pass through the linearized equation, because it is not well-posed at this low level of regularity.

This paper is concerned with the global dynamics of a two-species Grindrod clustering model with Lotka–Volterra competition. The model takes the advective flux to depend directly upon local population densities without requiring intermediate signals like attractants or repellents to form the aggregation so as to increase the chances of survival of individuals like human populations forming small nucleated settlements. By imposing appropriate boundary conditions, we establish the global boundedness of solutions in two-dimensional bounded domains. Moreover, we prove the global stability of spatially homogeneous steady states under appropriate conditions on system parameters, and show that the rate of convergence to the coexistence steady state is exponential while the rate of convergence to the competitive exclusion steady state is algebraic.

Abstract

In this paper we study the existence of bound states for the following class of quasilinear problems, $$begin{aligned} left{ begin{aligned}&-varepsilon ^pDelta _pu+V(x)u^{p-1}=f(u)+u^{p^*-1}, u>0, text {in} {mathbb {R}}^{N},&lim _{|x|rightarrow infty }u(x) = 0, end{aligned} right. end{aligned}$$ where (varepsilon >0) is small, (1<p<N,) f is a nonlinearity with general subcritical growth in the Sobolev sense, (p^{*} = pN/(N-p)) and V is a continuous nonnegative potential. By introducing a new set of hypotheses, our analysis includes the critical frequency case which allows the potential V to not be necessarily bounded below away from zero. We also study the regularity and behavior of positive solutions as (|x|rightarrow infty ) or (varepsilon rightarrow 0,) proving that they are uniformly bounded and concentrate around suitable points of ({mathbb {R}}^N,) that may include local minima of V.

In this paper, we study almost minimizers to a fractional Alt–Caffarelli–Friedman type functional. Our main results concern the optimal (C^{0,s}) regularity of almost minimizers as well as the structure of the free boundary. We first prove that the two free boundaries (F^+(u)=partial {u(cdot ,0)>0}) and (F^-(u)=partial {u(cdot ,0)<0}) cannot touch, that is, (F^+(u)cap F^-(u)=emptyset ). Lastly, we prove a flatness implies (C^{1,gamma }) result for the free boundary.

It is considered a semilinear elliptic partial differential equation in (mathbb {R}^N) with a potential that may vanish at infinity and a nonlinear term with subcritical growth. A positive solution is proved to exist depending on the interplay between the decay of the potential at infinity and the behavior of the nonlinear term at the origin. The proof is based on a penalization argument, variational methods, and (L^infty ) estimates. Those estimates allow dealing with settings where the nonlinear source may have supercritical, critical, or subcritical behavior near the origin. Results that provide the existence of multiple and infinitely many solutions when the nonlinear term is odd are also established.

In the present paper we prove that densely, with respect to an (L^p)-like topology, the Lyapunov exponents associated to linear continuous-time cocycles (Phi :mathbb {R}times Mrightarrow {{,textrm{GL},}}(2,mathbb {R})) induced by second order linear homogeneous differential equations (ddot{x}+alpha (varphi ^t(omega ))dot{x}+beta (varphi ^t(omega ))x=0) are almost everywhere distinct. The coefficients (alpha ,beta ) evolve along the (varphi ^t)-orbit for (omega in M) and (varphi ^t: Mrightarrow M) is an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation (ddot{x}+beta (varphi ^t(omega ))x=0) and for a Schrödinger equation (ddot{x}+(E-Q(varphi ^t(omega )))x=0), inducing a cocycle (Phi :mathbb {R}times Mrightarrow {{,textrm{SL},}}(2,mathbb {R})).

We study an overdetermined fully nonlinear problem driven by one of the Pucci’s Extremal Operators in an external domain. Under certain decay assumptions on the solution, we extend Serrin’s symmetry result, i.e, every domain where the solution exists must be radial.